Reviews of Claude Berge's books

We give below versions of reviews of four of Claude Berge's books. We have modified these to give more details of the mathematicians involved, inserting the full name of the mathematician where appropriate. It is interesting to note that graph theory was a new topic when Berge began to write on the subject and the reviews reflect this in that they assume that the reader may not have any idea about graph theory.

The tentacles of the theory of graphs steadily grow more numerous and penetrate more deeply into many phases of mathematics. The jargon of the theory includes such words and phrases as: arc, edge, path, chain, Hamiltonian circuit [named after William Rowan Hamilton], tree, node, function of Grundy [named after Patrick Michael Grundy (1917-1959)], latin square, incidence matrix, totally unimodular matrix, chromatic class, cyclomatic number, semi-factor, capacity, coupling, network. Applications of the theory spread over not only geometry, topology, and algebra, but also games, economics, military affairs, social structures, physics, statistics, communications, dynamic programming, operations research. Many famous questions which have helped to motivate the development of the broad theory of graphs are seemingly unrelated. To illustrate their diversity, we mention: the problem of the eight queens on the chessboard; the promenade of the fifteen school girls; generalizations of Nim; the boat ferrying the wolf, goat, and cabbage across the river one at a time; the transportation problem; the excursion problem (where m families with r1 ..., rm members respectively travel in n vehicles with s1 ..., sn available seats respectively, and no family has two of its members in the same conveyance); the assignment of personnel; the path of a knight over every square on the chessboard; the design of a tournament so that the results will be "fair"; the travelling salesman problem; the seven bridges of Königsberg; the longest circular sequence composed of zeroes and ones with no repeated portion of k consecutive digits; the number of days in a conference where eleven ministers sit at a round table but no two ministers sit beside each other more than one day; the most effective bombardment of communication channels; the non-crossing of supply lines for the three public utilities to the three cities; the dissection of a rectangular region into pair-wise incongruent square regions; the four-color problem. Of the several appendices, one lists fourteen questions still awaiting solution. All in all, in this book we have an up-to-date exposition, by one of the developers himself, of an intriguing theory capable of handling a fascinating potpourri of situations.

The term graph, the subject of this book, has not the common connotation of a plot or curve, but refers to an established but esoteric mathematical usage. A graph is a set of points, certain pairs of which are connected by arcs. These arcs may or may not be oriented, that is, have an associated definite direction from one endpoint to the other. Possibly a new name to dispel confusion would be apropos. We offer: theory of linkages. The geometric aspect of the above definition is of course not the heart of the matter; graphs can be symbolic diagrams of a rich variety of situations. The points can represent almost any kind of object and the arcs almost any type of interrelation between them. Thus we should expect the subject to abound in diverse applications, and Berge's book does. Thumbing through the volume, one is fascinated by the variegated array of illustrations, and the diverse examples attest to the very eclectic contents. The research analyst will find much to charm as well as instruct him. Hitherto the standard work was Denes König's 'Theorie der endlichen und unendlichen Graphen' (Leipzig, 1936). One read here of a branch of classical mathematics that struck out in many directions but penetrated deeply in few, the minor efforts of many major mathematicians. A purview of the subject can be had from a glance at a sampling of the more renowned problems. An Euler line [named after Leonhard Euler] is one that covers every arc of a graph and can be drawn without lifting the pencil or retracing. It stems from the famous problem of the seven bridges of Königsberg as recounted in most histories of mathematics and is popularly described as being the starting point of topology. A Hamilton line [named after William Rowan Hamilton] is nearly a dual concept: it need not cover all the arcs but must pass over each vertex once and only once. In both cases the problem is to ascertain conditions under which it is possible to draw the appropriate line. The Euler problem is rather easy, but the Hamilton problem is still unsolved. One of the most famous of all unsolved problems is that of map colouring: to show that four colours suffice to colour any planar map so that adjacent countries are always of distinct hue. It becomes a question of graph theory if we let each vertex represent a country, connecting them by an arc when the corresponding countries have a mutual border line. The cream of such older problems is elegantly treated in Berge's book. But along with them are the current advances in the subject. The disciple of operations research will recognize much material and many names; a good proportion is American. For example, there is a chapter on transportation networks. It includes the Ford-Fulkerson algorithms [named after Lester Randolph Ford (1886-1967), Delbert Ray Fulkerson (1924-1976)] and the theorems of Hoffman [Alan Jerome Hoffman (1924-)] and Gale [David Gale (1921-2008)]. As usual, the applications are astonishingly diverse. Besides questions of optimizing transport and routing, the same techniques are fitted to the problem of minimal covering, some combinatorial teasers, problems in set theory, and linear programming. The methods continue to solve problems in some subsequent chapters; in one on couplages we find a problem typical of the applied scope of this theory. In what are called simple graphs, the vertices are divided into two sets such that all arcs connect only members of one set with points of the other. A couplage is a subset of these arcs, with no two having a common endpoint. The problem: to find a maximum couplage. What is an application? Let one of the two sets of vertices above represent a set of workers, and the other, jobs to be done. A connecting arc is drawn when a worker is capable of fulfilling that job. Then a maximum couplage corresponds to a maximal scheme of assigning workers to suitable jobs. A second, but more trivial application, deserves to be quoted on less technical grounds:

There is a chapter on game theory (the author has done a separate monograph on this subject); others deal with matrices and trees. There is an appendix on electrical circuit theory and some allied non-electrical problems. Always there is stimulating diversity. In the chapter entitled Facteurs, for instance, three consecutive examples run: Voyage around the World (William Rowan Hamilton); The Knight's Tour of the Chessboard (Leonhard Euler); and - a quick plunge to modern times and queuing problems - The Bookbinding Problem [Selmer Martin Johnson (1916-1996)]. The operations analyst will benefit from this work (aside from enjoying himself) in acquiring an arsenal of new and imaginative techniques. Even should he never have occasion to use them, his wits cannot be but sharpened owing to the remarkable way seemingly disparate concepts are linked by common underlying ideas.

2. The Theory of Graphs and its Applications, by Claude Berge.

2.1. Review by: F Harary.Biometrika50 (1/2) (1963), 231.

The original book, written in French and published in 1958 by Dunod in Paris, was not reviewed in Biometrika. For a comprehensive discussion, see my comments in Math. Reviews21 (1960), 309. The English translation of this book has now appeared. Congratulations are in order to Alison Doig of the London School of Economics and Political Science on a most competent translating job. This is the second book on graph theory ever written. The previous book is already classical: Denes König, 'Theorie der endlichen und unendlichen Graphen' (Akademische Verlag, Leipzig, 1936; Chelsea Publishing Company, New York, 1950.) Recently a third book on the subject has appeared: 0ystein Ore, 'Theory of Graphs' (American Mathematical Society Colloquium Publications 38, 1962), and several other books are known to be in process. Of late there has been a resurgence of interest in both the theory and application of graphs, whence the author obtains the title of his book. The book contains a considerable number of new results in graph theory which have been discovered since the book of Denes König, and is therefore a most welcome addition to the mathematical literature. The English translation represents an improvement over the original book in French because of the corrections and omissions which have been made.

This is the English translation of "Théorie des graphes et ses applications," Dunod, Paris, 1958. Congratulations are in order to Alison Doig of the London School of Economics and Political Science on a most competent translating job. Occasionally cultural differences between the French and the British are noticeable, as in the Introduction where "II est tres remarquable . . . " is translated: "It is a matter of common observation .. " (that different disciplines often use analogous theorems). The French book was reviewed by R A Good in The American Mathematical Monthly68 (1961) 76-77. The first and last sentences of Good's review state: "The tentacles of the theory of graphs steadily grow more numerous and penetrate more deeply into many phases of mathematics. All in all, in this book we have an up-to-date exposition, by one of the developers himself, of an intriguing theory capable of handling a fascinating potpourri of situations."

In our review of the French book in Mathematical Reviews21 (1960), 309, we noted: "This is the second book on graph theory ever written. The previous book is the already classical: Denes König, 'Theorie der endlichen und unendlichen Graphen' (Akademische Verlag, Leipzig, 1936; Chelsea Publishing Company, New York, 1950). There are however several books on combinatorial analysis and topology which contain a chapter on graph theory. There has recently been a resurgence of interest in both the theory and application of graphs, whence the author obtains the title of his book. The book contains a considerable number of new results on graph theory which have been discovered since the book of Denes König, and is therefore a most welcome addition to the mathematical literature." The most noticeable change is that Appendices III, IV and V have been omitted in translation. Appendix IV in the original book stated 14 unsolved problems. Of these, Problem 4 was solved recently by Chong-Yun Chao, Problem 11 was solved in our previous review, and Problems 12-14 ask the reader to settle the Four-Colour-Conjecture. The accuracy of the references has been improved. Unfortunately some of them still refer to several papers. The inclusion of an author index in this English translation would have been very welcome. At the present time, the existing or announced books on graph theory are as follows: 1. Denes König, original in German, being translated into English. 2. Claude Berge, original in French, English translation herewith reviewed. 3. 0ystein Ore, Theory of Graphs, American Mathematical Society Colloquium Publications 38, 1962. 4. 0ystein Ore, Graphs and their uses, forthcoming in the School Mathematics Study Group (SMSG) series. In addition, several other graph theorists are actively engaged in writing their own versions of the fundamentals, foundations, and elements of graph theory. It is to be hoped that with all of these contributions to the field, as well as those books on graph theory written primarily for an audience of electrical engineers, operations researchers, or social scientists, two developments will become more pronounced: (i) each scholar who finds it convenient to use structural or combinatorial concepts in his own research will not feel obliged to rediscover graph theory for himself, ab initio. (ii) this elegant theory with its applications within mathematics to topology, logic, algebra, and combinatorial analysis will eventually become an undergraduate course at most modern universities.

2.3. Review by: E Burger.Econometrica33 (1) (1965), 251-252.

This is a translation of a French book previously published in 1958. Some changes from the original have been made. For a detailed review of the book from the point of view of pure mathematics compare Mathematical Reviews21, review 1608. Here, first of all, we must note the close connection existing between the theory of graphs and many fields of application. The author writes: "In this development of the theory of graphs, our aim has been to provide the reader with a mathematical tool which can be used in the behavioural sciences, in the theory of information, cybernetics, games, transport networks, etc." Practically, it does not presuppose much previous mathematical knowledge. Even the basic concepts of set theory and linear algebra are explained in the book. Only in chapters 14-16 is some elementary knowledge of matrices and determinants presupposed. The main feature, however, which makes the book suitable for the purpose stated above is the ample use of illuminating examples from the most diversified fields of application. For example, three chapters and one appendix of the book deal more or less with networks, one (at least) and one appendix deal with game theory, etc. On the whole, these examples show very clearly that the concepts and theorems of graph theory are fundamental for the investigation of many structures encountered in the real world, and that (as the author states) "The basic ideas [of the theory of graphs] are closely linked with reality, and can easily be identified in any specific case." Particularly will the behavioural scientist agree with this statement after reading the book. An indication of the wealth of material dealt with in this book can be gained from the following list of chapter headings (where we occasionally supply some supplementary indication about the applications treated in the corresponding chapter): 1. General definitions; 2. Descendance relations; 3. The ordinal function and the Grundy function [named after Patrick Michael Grundy (1917-1959)] on an infinite graph; 4 . The fundamental numbers of the theory of graphs (cyclotonic and chromatic number, coefficients of internal and external stability, application to information theory; 5. Kernels of a graph (application to game theory); 6. Games on a graph (Nim type games, games with perfect information); 7. The problem of the shortest route; 8. Transport networks; 9. The theorem of the demi-degrees; 10. Matching on a simple graph (assignment problem, Latin squares); 11. Factors (scheduling problems); 12. Centres of a graph; 13. Diameter of a strongly connected graph; 14. The matrix associated with a graph (psychological experiments; sociology: the problem of the leader); 15. Incidence matrices( electric relays); 16. Trees and arborescences (transport network, and Lukasiewicz's symbolism [named after Jan Lukasiewicz] without parentheses); 17. Euler's problem (telecommunications, cryptography); 18. Matching in the general case (switching circuits); 19. Semi-factors; 20. The connectivity of a graph (arranging a tournament); 21. Planar graphs (colouring problems); Appendix 1. Note on the general theory of games; Appendix 2. Note on transport problems.

The theory of graphs is in a rapid state of development at the present time and this book, being the first on the subject in English, is very much to be welcomed. [By the time this review appears, another will have been published and a third will be in the press. This latter is the first of two volumes which have arisen out of the projected 'Graph Theory' by Harary [Frank Harary (1921-2005)] and Norman [Robert Zane Norman (1924-)] announced by Claude Berge in the French (1958) edition of the book under review.] The word graph in the present context covers such things as electrical networks, family trees, sociograms or carbon chains in large molecules; the familiar "map" of the London Underground being a typical example which illustrates their diagrammatic nature. It is not, perhaps, a felicitous word, but its very vagueness does free it from particular associations and words like network are available for use in describing special types. In its full generality, a graph is merely a set of points which are commonly joined by lines but may be isolated, joined to themselves (by loops) and so on. The theory arose out of mathematical games and puzzles, and one meets in this book such old friends as: the Königsberg Bridges; the shortest path through a maze; Nim; the Ferryman, Wolf, Goat and Cabbage; the Knight's Tour; the Eight Queens and several others. Consonant with this the arguments are elementary but often far from easy. The problems treated are still somewhat heterogeneous and, whilst the basic subject matter and method give the book a natural unity, one gets the impression that the emphases on the different subdivisions of the subject are still to some extent the result of historical accident rather than measures of their importance to the theory as a whole. Of course, in a book of this sort, those parts which have so far found application will tend to be given more detailed coverage. Paradoxically it is the unified treatment that makes the book more difficult for the beginner; the first four chapters consist almost entirely of definitions and notation. On the other hand, only sixth-form mathematics is required plus elementary matrix theory for the later chapters. Statistical applications are by way of fringe benefits; to multiple ranking, experimental design, Markov chain and random walk theory, for example. More intimate relationships, discussed here, are with games theory and linear programming; the transport problem with its numerous Operations Research applications gets a quite detailed treatment. One may expect many more statistical applications in future since the problems of enumeration have been getting a great deal of successful attention recently and these are the ones which tend to arise statistically. Miss Alison Doig's translation is literal and very readable and it is no criticism (and perhaps only one man's opinion) to say the French passive seems better suited to impersonal prose than the English "we" and "one". She has cut two (already obsolete) appendices, extended slightly the index of terms used and made one or two minor alterations, but the book is substantially faithful to the original - too faithful perhaps; a full index would have been a welcome addition. So, too, would have been a table of equivalents; since different authors have different usages for concepts which, one day, will be described in standard terms but, so far, have many names.

3. Programming, Games and Transportation Networks, by Claude Berge and Alain Ghouila-Houri.

3.1. Review by: A M.Journal of the American Statistical Association62 (317) (1967), 309.

This book consists of two unrelated parts, Part I entitled, "General Theory of Convex Programming" by Alain Ghouila-Houri and Part II entitled, "Problems of Transportation and of Potential" by Claude Berge. The first part is a theoretical introduction to convex programming using topological properties of Euclidean space as an entree to the theory. The second part is a brief introduction to graph theory, network flow theory, and the transportation problem. A better reference for the material of this part is L R Ford and D R Fulkerson, "Flows in Networks," Princeton University Press, Princeton, 1962 [Lester Randolph Ford (1886-1967), Delbert Ray Fulkerson (1924-1976)].

3.2. Review by: S Vajda.The Mathematical Gazette50 (373) (1966), 347.

This is another item in the continuing flow of translations of English titles into French and vice versa. The book starts with the contribution of the second named author, who discusses the "general theory of convex programming," i.e. of finding the minimum of a convex function, subject to inequalities. In fact, he supplies the preliminary mathematical concepts, such as sets, vector space, convexity of sets and of functions, and proves the Minimax Theorem of John von Neumann and its generalization by Maurice Sion (who replaces convexity in von Neumann's theorem by quasi-convexity), as well as the Farkas-Minkowski theorem [named after Gyula Farkas and Hermann Minkowski]. As regards the general theory of convex programming, only about ten pages are devoted to the fundamental Kuhn-Tucker theory [named after Harold William Kuhn (1925-) and Albert William Tucker], and the last two sections treat convex programmes with linear constraints, and games of strategy. Computational routines are merely hinted at, in an unsystematic fashion, and could not be learned from this text. Altogether, this is a well-written textbook, in the French manner, which might be of interest on this side of the Channel to those who teach the subject and welcome differently tinted spectacles, for a change. The second part, by Claude Berge, is a reduced and possibly brought up-to-date version of an earlier book (Theory of Graphs and its Applications, translated by Alison Doig). The Preface tells us that it "aims to demonstrate that a large part of these mathematical programming problems can be solved in a simple and elegant manner using ... the theory of graphs". The problems which can be thus solved are, naturally, of a combinatorial character and are akin to puzzles and to recreational mathematics, and the author makes ample use of his opportunities to be instructive as well as entertaining. The purist will occasionally be shocked, but one reader at least is glad to have a source book of a theoretical treatment of such well-known problems as the "wolf, the cabbage and the goat" (similar to the "cannibals and missionaries"), getting out of a maze, or dissecting squares into smaller squares. All this is material for mathematics without tears, though the rigour of presentation is sometimes more apparent than real. The diagrams are excellent and relieve the aspect of pages full of symbols. It is also pleasant to note that the paper is much better than that of the original French edition.

3.3. Review by: Adi Ben-Israel.SIAM Review10 (2) (1968), 238-239.

This translation of 'Programmes, jeux et reseaux de transport', Dunod, Paris, 1962 (reviewed in Mathematical Reviews 33 (1967), No 1137) is a valuable addition to the literature of mathematical programming. It is an exposition of the analytical aspects of mathematical programming, and its underlying mathematical structures, which are of independent interest as well as relevant to other areas of applied mathematics. This book is essentially self-contained, requiring but a casual knowledge of linear algebra and advanced calculus. Tuned for mathematicians, it should be equally useful for engineers and scientists as a well-motivated introduction to modern analysis. This book is not concerned with the algorithmic and computational aspects of mathematical programming, nor with some other parts of the usual mathematical programming curriculum, e.g., integer programming, dynamic programming. In spite of this (and the lack of exercises and an adequate bibliography for Part 1) it may be used, in conjunction with a more conventional text, as a textbook in mathematical programming for beginning graduate students. The book is well organized, clearly written and, to this reviewer, a pleasure to read. The translation and printing are excellent. The book is divided into two (almost independent) parts, written respectively in the spirit of two well-known books by the first author: 'Topological Spaces', Macmillan, New York, 1963 (Mathematical Reviews 21 (1960), No 4401) and 'The Theory of Graphs and its Applications', John Wiley, New York, 1962 (Mathematical Reviews 21 (1960), No 1608). Part 1, 'General Theory of Convex Programming', by Alain Ghouila-Houri, is true to its name. The algebraic and topological prerequisites are adequately presented in Chapters 1 and 2 (50 pages). Chapter 3, Properties of Convex Sets and Functions in the Space Rn (18 pages), discusses (in Rn) the following topics: Separation of convex sets, supporting hyperplanes, the Krein-Milman theorem [named after Mark Grigorievich Krein and David Pinhusovich Milman (1912-1982)], intersections of convex sets, Helly's theorem [named after Eduard Helly], minimax theorems and several generalizations of the Farkas-Minkowski theorem [named after Gyula Farkas and Hermann Minkowski]. (The one on page 67 requires an additional hypothesis such as: "f (x) > 0 for some x in Rn" in order to exclude the case: p0 = 1, yi = 0, i = 1, ... , m.) Chapter 4, Programming and Associated Problems (14 pages), develops the Kuhn-Tucker theory [named after Harold William Kuhn (1925-) and Albert William Tucker] for convex programming. The latter, under linear constraints, is the topic of Chapter 5 (17 pages), where also the simplex and Frank-Wolfe algorithms [named after Marguerite Josephine Straus Frank and Philip Wolfe] are outlined, and duality is developed by using conjugate functions (no reference to W Fenchel [Moritz Werner Fenchel (1905-1988)], On conjugate convex functions, Canad. J. Math. 1 (1949), 73-77; Convex cones, sets and functions, Mimeographed lecture notes, Princeton University, 1953). Chapter 6, Games of Strategy (12 pages), hardly justifies the word "Games" in the book's title. 2-person 0-sum games are introduced and solved by linear programming and by successive approximations (without mentioning George W Brown, Iterative solution of games by fictitious play, Activity Analysis of Production and Allocation, T C Koopmans [Tjalling Charles Koopmans (1910-1985)], ed., John Wiley, New York, 1951, 374-376; Julia Robinson, An iterative method for solving a game, Ann. Math. 54 (1951), 296-301). Part II, 'Problems of Transportation and of Potential' by Claude Berge, treats such problems by methods of graph theory, whose prerequisites are given in Chapter 7, Cycles and Coboundaries of a Graph (24 pages). Flows, potentials and the duality relating them are studied in Chapter 8 (29 pages). Chapter 9, Flow Algorithms (43 pages), presents the main algorithms for solving extremum problems of paths, spanning trees, potential differences and flows. Chapter 10, Problems Related to the Transportation Problem (25 pages), contains some very interesting applications, e.g., multi-terminal problems and multi-commodity networks.

This book is an English translation by Maxine Merrington and C Ramanujacharyulu of the book 'Programmes, jeux, et réseaux de transport' by the same authors published in 1962 and reviewed in Operations Research11 (1963), 286. This new version appears to be a very close translation. Subsequent reflection on our earlier comments leads to one revised opinion: We regret mentioning a bifurcated book. Better to speak of two treatments of the same subject from different viewpoints. Mastery of this kind of diversity is one of the best ways really to understand a topic, and presentation of the first and third items in the title in such a double manner makes this an unusually fine book. It is fair to warn the reader whose prime interest is games that this is not the book for him. There is but one relevant chapter that covers only the rudiments of finite, matrix games. It is there because its treatment appends nicely to the main business of the book rather than of its intrinsic interest.

This book first appeared in the French edition of 1962. It is in reality two books of roughly equal length. For, although the two parts, respectively by Alain Ghouila-Houri and by Claude Berge, both deal with series of problems which cover much of the mathematical theory of the non-statistical part of operational research and both sets of problems are solved by the construction of algorithms of linear programming type (with proof of convergence), yet their mathematical nature is very different. Part I, called the "General Theory of Convex Programming", itself falls into two parts. The first half runs through the elements of set theory, vector spaces and the topological properties of Rn. This is a very useful brief introduction to these topics that is not very strongly oriented to the second half, which comprises, in the main, three chapters. These are: one on convex sets and functions in Rn,one on minimization of functions subject to differentiable conditions (inequalities and constraints) and one where the functions to be minimized are convex and the constraints linear. Their contents are respectively: separation theorems for convex sets, the Farkas-Minkowski [named after Gyula Farkas and Hermann Minkowski] and von Neumann minimax theorems, with divers extensions and corollaries; the various forms of the minimization problem (with equivalence theorems); algorithms of "simplex" type for the solution of convex and quadratic programming problems. Finally, there is a short chapter on the connection between theory of games and programming.

Part II, is concerned with the study of flows and potential differences in networks. A chapter on the underlying graph theory is followed by one giving a general discussion of formulation and what might be called structural theorems on flows (transportation) and potential differences. These lay the basis for the main chapter which gives both theorems (such as the max. flow - min. cut) and algorithms for, inter alia, the path problem, shortest paths, minimal spanning tree and divers maximum flow problems. Finally, a chapter, is devoted to generalizations and specializations of the maximum flow problem. Whilst Part II is made more instructive (and more readable) by sketches of the operational research set-ups which gave rise to the problems, both parts are equally shy of computer-programming and numerical considerations. Rates of convergence or even computability in terms of existing computers are not discussed. Although there is a certain amount of reference to papers on the more practical aspects, there is nothing like the coverage given, for instance, in Ford and Fulkerson's (1962) book which deals with much the same field as Berge does. Surprisingly, Vajda's excellent 'Mathematical Programming' (1961) [Steven Vajda (1901-1995)] is not mentioned, though in that book all the essential ideas of the present one are explained in simple numerical terms so that this one could almost have been subtitled "A Mathematical Gloss on a Text by Vajda". Since both parts of the present book are concerned with rigorous and often intricate chains of mathematical propositions (it is very much modern mathematics) from a variety of disciplines, one must remark that the translation is uncommonly meticulous. Great care has been taken by Maxine Merrington, to whom we owe the present translation, in obtaining the precise English equivalent for each of the multitude of technical mathematical terms and she acknowledges her good fortune in having had available expert advice from her colleagues at University College in a wide range of fields. It would have been intolerable if not impossible, to read the book without this precision.

4. The General Theory of n-Person Games, by Claude Berge.

4.1. Review by: Martin Shubik.Econometrica29 (4) (1961), 821.

This brief book (114 pages) is primarily of interest to the mathematician. The argument is presented in a highly abstract manner and no consideration is given to applications to economics. Berge's original contribution comes in the application of the methods of graph theory to extensive form games. Although the book will be of little direct interest to most economists, both it and the recently published two volumes of Samuel Karlin serve to indicate the quantity and quality of serious advanced mathematical work required for the rigorous exploration of detailed models of behaviour.

JOC/EFR January 2013

The URL of this page is:
https://www-history.mcs.st-andrews.ac.uk/Extras/Berge_books.html